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arXiv · 2205.05364

An effective criterion for Nielsen-Schreier varieties

Vladimir Dotsenko·Ualbai Umirbaev
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Abstract

All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen-Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen-Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen-Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen-Schreier varieties of algebras with one binary operation and identities of degree four.

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Vladimir Dotsenko, Ualbai Umirbaev. 2022-05-11. An effective criterion for Nielsen-Schreier varieties. https://doi.org/10.1093/imrn%2Frnad092

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